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Chaos, Drawn

Chapter 3Weather & the butterfly

In 1961 a weather researcher rounded three numbers off at the third decimal to save paper, ran the forecast again, and got a different season. That accident is why the ten-day forecast will never come.

Three equations for a sky

Edward Lorenz stripped a weather model down to three numbers — how fast the air rolls, and two about how the heat sits — and three rules for how they push each other.5 No noise, no dice. He expected it to settle or to loop. It did neither.

ẋ = σ(y − x)    ẏ = x(ρ − z) − y    ż = xy − βz

Reading it: the little dot means “rate of change”. Each line says how fast one number moves given where all three are now. Feed in a start, take tiny steps, and the three numbers trace a path through the air.

-20-100102001020304050xzThe Lorenz attractor · σ=10, ρ=28, β=8/3
Fig 5. The path those three rules draw, with Lorenz's own settings. It circles one wing, jumps to the other, circles a few times, jumps back — the order of the jumps never repeating. This shape is a strange attractor: every start is pulled onto it, and no start ever settles down on it. · computed here · Nan · hongdam.net · CC BY 4.0

Round off at the third decimal, lose the month

Here is Lorenz's accident, on purpose. Two runs of the same three equations, one started a hundred-thousandth away from the other in a single number.

051015202530-20-1001020timex (the state)Same rule, two starts 0.00001 apartthe two forecasts part
Fig 6. For a good while the two forecasts are one line — you could not slip a knife between them. Then they part, and within a day or two one says storm and the other says clear. Same equations. Same knob. A rounding error at the start. · computed here · Nan · hongdam.net · CC BY 4.0

How far ahead can anyone see?

This is the part you can actually measure. Track the gap between the two runs and plot it on a scale where each step up is ten times bigger. The gap climbs a straight line — meaning it multiplies by a fixed factor every second — until it is as wide as the whole butterfly and cannot grow more.

1101e-091e-081e-071e-061e-050.00010.0010.010.1110100timetypical gapThe gap grows by a fixed factor each secondslope λ ≈ 0.92 per secondthe size of the whole butterfly — the gap can't grow past it
Fig 7. The gap grows by a steady factor, the same factor every second. The slope of that climb is the Lyapunov exponent, λ ≈ 0.92 per second here. It is the single number that says how chaotic a system is. · computed here · Nan · hongdam.net · CC BY 4.0
gap(t) ≈ gap(0) · eλt

Reading it: the gap you start with, times a number that doubles and redoubles on the clock. With λ ≈ 0.92, the gap grows about 2.5-fold every second of model time.

Turn it around and you get the forecast horizon. To keep the gap useful you must start with it this much smaller — so every extra decimal of accuracy in your measurements buys you only a fixed stretch more warning, not a proportional one. Ten times better instruments push the horizon out by the same short step as the ten times before them. On this model that step is about 1.1 seconds of model time per factor of ten.

Why the weather app stops at ten days

The real atmosphere has a doubling time for small errors of roughly a day and a half. Start with the best measurements on Earth and the unknown in them still grows to swamp the forecast in about two weeks. Better satellites help — but only by that same fixed step each time. Two weeks is not a budget problem. It is the Lyapunov wall.6